In this paper, the flexural free vibrations of three dimensional micro beams are investigated based on strain gradient theory. The most general form of the strain gradient theory which contains five higher-order material constants has been applied to the micro beam to take the small-scale effects into account. Having considered the Euler-Bernoulli beam model, governing equations of motion are w...

We consider the cubic nonlinear fourth-order Schrödinger equation \begin{document}$ i \partial_t u - \Delta^2 + \mu \Delta = \pm |u|^2 u, \quad \geq 0 $\end{document} on $ \mathbb R^N, N\geq 5 with random initial data. prove almost sure local well-posedness below scaling critical regularity. also probabilistic small data global and scattering. Finally, we scattering a large probability for rand...

Abstract The fourth-order discrete Dirichlet boundary value problem is also a elastic beam problem. In this paper, the existence of infinitely many solutions to investigated through critical point theory. By an important inequality we established and oscillatory behavior f either near origin or at infinity, obtain solutions, which converge zero unbounded. end, two examples are presented illustr...

Understanding of space charge effects has grown and recent studies have led to the findings of space charge resonances in high intensity linear accelerators. Lately the sixth order resonance of high intensity linear accelerators was reported, along with the in-depth studies on the interplay of the fourth order resonance and the envelope instability. Experiment studies on space charge resonances...